package strconv
-import "math"
-
// An extFloat represents an extended floating-point number, with more
// precision than a float64. It does not try to save bits: the
// number represented by the structure is mant*(2^exp), with a negative
// floatBits returns the bits of the float64 that best approximates
// the extFloat passed as receiver. Overflow is set to true if
// the resulting float64 is ±Inf.
-func (f *extFloat) floatBits() (bits uint64, overflow bool) {
- flt := &float64info
+func (f *extFloat) floatBits(flt *floatInfo) (bits uint64, overflow bool) {
f.Normalize()
exp := f.exp + 63
exp += n
}
- // Extract 1+flt.mantbits bits.
+ // Extract 1+flt.mantbits bits from the 64-bit mantissa.
mant := f.mant >> (63 - flt.mantbits)
if f.mant&(1<<(62-flt.mantbits)) != 0 {
// Round up.
// Infinities.
if exp-flt.bias >= 1<<flt.expbits-1 {
- goto overflow
- }
-
- // Denormalized?
- if mant&(1<<flt.mantbits) == 0 {
+ // ±Inf
+ mant = 0
+ exp = 1<<flt.expbits - 1 + flt.bias
+ overflow = true
+ } else if mant&(1<<flt.mantbits) == 0 {
+ // Denormalized?
exp = flt.bias
}
- goto out
-
-overflow:
- // ±Inf
- mant = 0
- exp = 1<<flt.expbits - 1 + flt.bias
- overflow = true
-
-out:
// Assemble bits.
bits = mant & (uint64(1)<<flt.mantbits - 1)
bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
return
}
-// Assign sets f to the value of x.
-func (f *extFloat) Assign(x float64) {
- if x < 0 {
- x = -x
- f.neg = true
- }
- x, f.exp = math.Frexp(x)
- f.mant = uint64(x * float64(1<<64))
- f.exp -= 64
-}
-
-// AssignComputeBounds sets f to the value of x and returns
+// AssignComputeBounds sets f to the floating point value
+// defined by mant, exp and precision given by flt. It returns
// lower, upper such that any number in the closed interval
-// [lower, upper] is converted back to x.
-func (f *extFloat) AssignComputeBounds(x float64) (lower, upper extFloat) {
- // Special cases.
- bits := math.Float64bits(x)
- flt := &float64info
- neg := bits>>(flt.expbits+flt.mantbits) != 0
- expBiased := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
- mant := bits & (uint64(1)<<flt.mantbits - 1)
-
- if expBiased == 0 {
- // denormalized.
- f.mant = mant
- f.exp = 1 + flt.bias - int(flt.mantbits)
- } else {
- f.mant = mant | 1<<flt.mantbits
- f.exp = expBiased + flt.bias - int(flt.mantbits)
- }
+// [lower, upper] is converted back to the same floating point number.
+func (f *extFloat) AssignComputeBounds(mant uint64, exp int, neg bool, flt *floatInfo) (lower, upper extFloat) {
+ f.mant = mant
+ f.exp = exp - int(flt.mantbits)
f.neg = neg
+ if f.exp <= 0 && mant == (mant>>uint(-f.exp))<<uint(-f.exp) {
+ // An exact integer
+ f.mant >>= uint(-f.exp)
+ f.exp = 0
+ return *f, *f
+ }
+ expBiased := exp - flt.bias
upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg}
- if mant != 0 || expBiased == 1 {
+ if mant != 1<<flt.mantbits || expBiased == 1 {
lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg}
} else {
lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg}
// Normalize normalizes f so that the highest bit of the mantissa is
// set, and returns the number by which the mantissa was left-shifted.
-func (f *extFloat) Normalize() uint {
- if f.mant == 0 {
+func (f *extFloat) Normalize() (shift uint) {
+ mant, exp := f.mant, f.exp
+ if mant == 0 {
return 0
}
- exp_before := f.exp
- for f.mant < (1 << 55) {
- f.mant <<= 8
- f.exp -= 8
+ if mant>>(64-32) == 0 {
+ mant <<= 32
+ exp -= 32
+ }
+ if mant>>(64-16) == 0 {
+ mant <<= 16
+ exp -= 16
}
- for f.mant < (1 << 63) {
- f.mant <<= 1
- f.exp -= 1
+ if mant>>(64-8) == 0 {
+ mant <<= 8
+ exp -= 8
}
- return uint(exp_before - f.exp)
+ if mant>>(64-4) == 0 {
+ mant <<= 4
+ exp -= 4
+ }
+ if mant>>(64-2) == 0 {
+ mant <<= 2
+ exp -= 2
+ }
+ if mant>>(64-1) == 0 {
+ mant <<= 1
+ exp -= 1
+ }
+ shift = uint(f.exp - exp)
+ f.mant, f.exp = mant, exp
+ return
}
// Multiply sets f to the product f*g: the result is correctly rounded,
1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
}
-// AssignDecimal sets f to an approximate value of the decimal d. It
+// AssignDecimal sets f to an approximate value mantissa*10^exp. It
// returns true if the value represented by f is guaranteed to be the
-// best approximation of d after being rounded to a float64.
-func (f *extFloat) AssignDecimal(d *decimal) (ok bool) {
+// best approximation of d after being rounded to a float64 or
+// float32 depending on flt.
+func (f *extFloat) AssignDecimal(mantissa uint64, exp10 int, neg bool, trunc bool, flt *floatInfo) (ok bool) {
const uint64digits = 19
const errorscale = 8
- mant10, digits := d.atou64()
- exp10 := d.dp - digits
errors := 0 // An upper bound for error, computed in errorscale*ulp.
-
- if digits < d.nd {
+ if trunc {
// the decimal number was truncated.
errors += errorscale / 2
}
- f.mant = mant10
+ f.mant = mantissa
f.exp = 0
- f.neg = d.neg
+ f.neg = neg
// Multiply by powers of ten.
i := (exp10 - firstPowerOfTen) / stepPowerOfTen
adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen
// We multiply by exp%step
- if digits+adjExp <= uint64digits {
- // We can multiply the mantissa
- f.mant *= uint64(float64pow10[adjExp])
+ if adjExp < uint64digits && mantissa < uint64pow10[uint64digits-adjExp] {
+ // We can multiply the mantissa exactly.
+ f.mant *= uint64pow10[adjExp]
f.Normalize()
} else {
f.Normalize()
// The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits.
//
// In many cases the approximation will be good enough.
- const denormalExp = -1023 - 63
- flt := &float64info
+ denormalExp := flt.bias - 63
var extrabits uint
if f.exp <= denormalExp {
+ // f.mant * 2^f.exp is smaller than 2^(flt.bias+1).
extrabits = uint(63 - flt.mantbits + 1 + uint(denormalExp-f.exp))
} else {
extrabits = uint(63 - flt.mantbits)
// f by an approximate power of ten 10^-exp, and returns exp10, so
// that f*10^exp10 has the same value as the old f, up to an ulp,
// as well as the index of 10^-exp in the powersOfTen table.
-// The arguments expMin and expMax constrain the final value of the
-// binary exponent of f.
-func (f *extFloat) frexp10(expMin, expMax int) (exp10, index int) {
- // it is illegal to call this function with a too restrictive exponent range.
- if expMax-expMin <= 25 {
- panic("strconv: invalid exponent range")
- }
+func (f *extFloat) frexp10() (exp10, index int) {
+ // The constants expMin and expMax constrain the final value of the
+ // binary exponent of f. We want a small integral part in the result
+ // because finding digits of an integer requires divisions, whereas
+ // digits of the fractional part can be found by repeatedly multiplying
+ // by 10.
+ const expMin = -60
+ const expMax = -32
// Find power of ten such that x * 10^n has a binary exponent
- // between expMin and expMax
- approxExp10 := -(f.exp + 100) * 28 / 93 // log(10)/log(2) is close to 93/28.
+ // between expMin and expMax.
+ approxExp10 := ((expMin+expMax)/2 - f.exp) * 28 / 93 // log(10)/log(2) is close to 93/28.
i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen
Loop:
for {
}
// frexp10Many applies a common shift by a power of ten to a, b, c.
-func frexp10Many(expMin, expMax int, a, b, c *extFloat) (exp10 int) {
- exp10, i := c.frexp10(expMin, expMax)
+func frexp10Many(a, b, c *extFloat) (exp10 int) {
+ exp10, i := c.frexp10()
a.Multiply(powersOfTen[i])
b.Multiply(powersOfTen[i])
return
}
+// FixedDecimal stores in d the first n significant digits
+// of the decimal representation of f. It returns false
+// if it cannot be sure of the answer.
+func (f *extFloat) FixedDecimal(d *decimalSlice, n int) bool {
+ if f.mant == 0 {
+ d.nd = 0
+ d.dp = 0
+ d.neg = f.neg
+ return true
+ }
+ if n == 0 {
+ panic("strconv: internal error: extFloat.FixedDecimal called with n == 0")
+ }
+ // Multiply by an appropriate power of ten to have a reasonable
+ // number to process.
+ f.Normalize()
+ exp10, _ := f.frexp10()
+
+ shift := uint(-f.exp)
+ integer := uint32(f.mant >> shift)
+ fraction := f.mant - (uint64(integer) << shift)
+ ε := uint64(1) // ε is the uncertainty we have on the mantissa of f.
+
+ // Write exactly n digits to d.
+ needed := n // how many digits are left to write.
+ integerDigits := 0 // the number of decimal digits of integer.
+ pow10 := uint64(1) // the power of ten by which f was scaled.
+ for i, pow := 0, uint64(1); i < 20; i++ {
+ if pow > uint64(integer) {
+ integerDigits = i
+ break
+ }
+ pow *= 10
+ }
+ rest := integer
+ if integerDigits > needed {
+ // the integral part is already large, trim the last digits.
+ pow10 = uint64pow10[integerDigits-needed]
+ integer /= uint32(pow10)
+ rest -= integer * uint32(pow10)
+ } else {
+ rest = 0
+ }
+
+ // Write the digits of integer: the digits of rest are omitted.
+ var buf [32]byte
+ pos := len(buf)
+ for v := integer; v > 0; {
+ v1 := v / 10
+ v -= 10 * v1
+ pos--
+ buf[pos] = byte(v + '0')
+ v = v1
+ }
+ for i := pos; i < len(buf); i++ {
+ d.d[i-pos] = buf[i]
+ }
+ nd := len(buf) - pos
+ d.nd = nd
+ d.dp = integerDigits + exp10
+ needed -= nd
+
+ if needed > 0 {
+ if rest != 0 || pow10 != 1 {
+ panic("strconv: internal error, rest != 0 but needed > 0")
+ }
+ // Emit digits for the fractional part. Each time, 10*fraction
+ // fits in a uint64 without overflow.
+ for needed > 0 {
+ fraction *= 10
+ ε *= 10 // the uncertainty scales as we multiply by ten.
+ if 2*ε > 1<<shift {
+ // the error is so large it could modify which digit to write, abort.
+ return false
+ }
+ digit := fraction >> shift
+ d.d[nd] = byte(digit + '0')
+ fraction -= digit << shift
+ nd++
+ needed--
+ }
+ d.nd = nd
+ }
+
+ // We have written a truncation of f (a numerator / 10^d.dp). The remaining part
+ // can be interpreted as a small number (< 1) to be added to the last digit of the
+ // numerator.
+ //
+ // If rest > 0, the amount is:
+ // (rest<<shift | fraction) / (pow10 << shift)
+ // fraction being known with a ±ε uncertainty.
+ // The fact that n > 0 guarantees that pow10 << shift does not overflow a uint64.
+ //
+ // If rest = 0, pow10 == 1 and the amount is
+ // fraction / (1 << shift)
+ // fraction being known with a ±ε uncertainty.
+ //
+ // We pass this information to the rounding routine for adjustment.
+
+ ok := adjustLastDigitFixed(d, uint64(rest)<<shift|fraction, pow10, shift, ε)
+ if !ok {
+ return false
+ }
+ // Trim trailing zeros.
+ for i := d.nd - 1; i >= 0; i-- {
+ if d.d[i] != '0' {
+ d.nd = i + 1
+ break
+ }
+ }
+ return true
+}
+
+// adjustLastDigitFixed assumes d contains the representation of the integral part
+// of some number, whose fractional part is num / (den << shift). The numerator
+// num is only known up to an uncertainty of size ε, assumed to be less than
+// (den << shift)/2.
+//
+// It will increase the last digit by one to account for correct rounding, typically
+// when the fractional part is greater than 1/2, and will return false if ε is such
+// that no correct answer can be given.
+func adjustLastDigitFixed(d *decimalSlice, num, den uint64, shift uint, ε uint64) bool {
+ if num > den<<shift {
+ panic("strconv: num > den<<shift in adjustLastDigitFixed")
+ }
+ if 2*ε > den<<shift {
+ panic("strconv: ε > (den<<shift)/2")
+ }
+ if 2*(num+ε) < den<<shift {
+ return true
+ }
+ if 2*(num-ε) > den<<shift {
+ // increment d by 1.
+ i := d.nd - 1
+ for ; i >= 0; i-- {
+ if d.d[i] == '9' {
+ d.nd--
+ } else {
+ break
+ }
+ }
+ if i < 0 {
+ d.d[0] = '1'
+ d.nd = 1
+ d.dp++
+ } else {
+ d.d[i]++
+ }
+ return true
+ }
+ return false
+}
+
// ShortestDecimal stores in d the shortest decimal representation of f
// which belongs to the open interval (lower, upper), where f is supposed
// to lie. It returns false whenever the result is unsure. The implementation
// uses the Grisu3 algorithm.
-func (f *extFloat) ShortestDecimal(d *decimal, lower, upper *extFloat) bool {
+func (f *extFloat) ShortestDecimal(d *decimalSlice, lower, upper *extFloat) bool {
if f.mant == 0 {
- d.d[0] = '0'
- d.nd = 1
+ d.nd = 0
d.dp = 0
d.neg = f.neg
+ return true
+ }
+ if f.exp == 0 && *lower == *f && *lower == *upper {
+ // an exact integer.
+ var buf [24]byte
+ n := len(buf) - 1
+ for v := f.mant; v > 0; {
+ v1 := v / 10
+ v -= 10 * v1
+ buf[n] = byte(v + '0')
+ n--
+ v = v1
+ }
+ nd := len(buf) - n - 1
+ for i := 0; i < nd; i++ {
+ d.d[i] = buf[n+1+i]
+ }
+ d.nd, d.dp = nd, nd
+ for d.nd > 0 && d.d[d.nd-1] == '0' {
+ d.nd--
+ }
+ if d.nd == 0 {
+ d.dp = 0
+ }
+ d.neg = f.neg
+ return true
}
- const minExp = -60
- const maxExp = -32
upper.Normalize()
// Uniformize exponents.
if f.exp > upper.exp {
lower.exp = upper.exp
}
- exp10 := frexp10Many(minExp, maxExp, lower, f, upper)
+ exp10 := frexp10Many(lower, f, upper)
// Take a safety margin due to rounding in frexp10Many, but we lose precision.
upper.mant++
lower.mant--
// Count integral digits: there are at most 10.
var integerDigits int
- for i, pow := range uint64pow10 {
- if uint64(integer) >= pow {
- integerDigits = i + 1
+ for i, pow := 0, uint64(1); i < 20; i++ {
+ if pow > uint64(integer) {
+ integerDigits = i
+ break
}
+ pow *= 10
}
for i := 0; i < integerDigits; i++ {
pow := uint64pow10[integerDigits-i-1]
return false
}
-// adjustLastDigit modifies d = x-currentDiff*ε, to get closest to
+// adjustLastDigit modifies d = x-currentDiff*ε, to get closest to
// d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε.
// It assumes that a decimal digit is worth ulpDecimal*ε, and that
// all data is known with a error estimate of ulpBinary*ε.
-func adjustLastDigit(d *decimal, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool {
+func adjustLastDigit(d *decimalSlice, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool {
if ulpDecimal < 2*ulpBinary {
// Approximation is too wide.
return false
projects / platform / upstream / gcc48.git / blobdiff